Appendix The Trigonometric Identity

MO (molecular orbital) Bloch orbital (crystal orbital) 

MO models with electron repulsion Hiickel-Hubbard Hamiltonian 

To end this section, it may be useful to the reader to give a table collecting some analogies between molecular and solid state theory (Table 3.4). The table is taken from Albright et al. (1985), and is useful in connecting quantum theorist terminology to that of solid state physicists. 

The trigonometric identity (Equation 3.62) can be easily derived as follows. We start from the well-known trigonometric formulae  

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For an arbitrary choice of co the left-hand side is still an oscillating function of time, while the right-hand side is a constant. The only way this equation can work for all t is if the two terms in the bracket give the same sum for any value of t. Remembering the trigonometric identity (see the end note to Appendix 9)... 

Calculate the first-order perturbation correction to the ground-state energy level using the particle in a box with V(x) = 0 for 0 x a, V(x) = oo for x < 0, x > a as the unperturbed system. Then calculate the first-order perturbation correction to the ground-state wave function, terminating the expansion after the term k = 5. (See Appendix A for trigonometric identities and integrals.)... 

Verify the derivation of Eq. 12.6 and 12.7 by using trigonometric identities from Appendix C. 

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